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1 Parametric polymorphism

Parametric polymorphism refers to when the type of a value contains type variables (denoted in Haskell by names in types starting with a lowercase letter) with no constraints (e.g. class constraints; see below) on them. Then the value may adopt any type that results from substituting those variables with concrete types.

In languages other than Haskell, this feature is often called generics.

For example, the function

id:: a -> a

contains an unconstrained type variable

a

in its type, and so can be used in a context requiring

Char->Char

or

Integer->Integer

or

(Bool->MaybeBool)->(Bool->MaybeBool)

or any of a literally infinite list of other possibilities. Likewise, the empty list

[]::[a]

belongs to every list type, and the polymorphic function

map::(a -> b)->[a]->[b]

may operate on any function type. Note, however, that if a single type variable appears multiple times, it must take the same type everywhere it appears, so e.g. the result type of

id

must be the same as the argument type, and the input and output types of the function given to

map

must match up with the list types.

Since a parametrically polymorphic value does not "know" anything about the unconstrained type variables, it must behave the same regardless of its type. This is a somewhat limiting but extremely useful property known as parametricity.

2 Ad-hoc polymorphism

Ad-hoc polymorphism refers to when a value is able to adopt any one of a finite number of types because it, or a value it uses, has been given a separate definition for each of those types. In many languages this is called overloading, and in Haskell it is achieved via the system of type classes and class instances.

Despite the similarity of the name, Haskell's type classes are quite different from the classes of most object-oriented languages. They have more in common with interfaces, in that they specify a series of methods or values by their type signature, to be implemented by an instance declaration.

So, for example, if my type can be compared for equality (most types can, but some, particularly function types, cannot) then I can give an instance declaration of the

Eq

class. All I have to do is specify the behaviour of the

==

operator on my type, and I gain the ability to use all sorts of functions defined using that operator, e.g. checking if a value of my type is present in a list, or looking up a corresponding value in a list of pairs.
You can recognise the presence of ad-hoc polymorphism by looking for constrained type variables: that is, variables that appear to the left of

=>

, like in

elem::(Eq a)=> a ->[a]->Bool

. Note that

lookup::(Eq a)=> a ->[(a,b)]->Maybe b

exhibits both parametric (in

b

) and ad-hoc (in

a

) polymorphism.

3 Other kinds of polymorphism

There are some kinds of polymorphism that Haskell doesn't support, or at least not natively, e.g. inclusion polymorphism and subtyping, common in OO languages, where values of one type can act as values of another type.